In statistics the Maxwell–Boltzmann distribution is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and used in physics (in particular in statistical mechanics) for describing particle speeds in idealized gases where the particles move freely inside a stationary container without interacting with one another, except for very brief collisions
in which they exchange energy and momentum with each other or with
their thermal environment. Particle in this context refers to gaseous
particles (atoms or molecules), and the system of particles is assumed to have reached thermodynamic equilibrium.[1] While the distribution was first derived by Maxwell in 1860 on heuristic grounds,[2] Boltzmann later carried out significant investigations into the physical origins of this distribution.

A particle speed probability distribution indicates which speeds are
more likely: a particle will have a speed selected randomly from the
distribution, and is more likely to be within one range of speeds than
another. The distribution depends on the temperature of the system and the mass of the particle.[3] The Maxwell–Boltzmann distribution applies to the classical ideal gas, which is an idealization of real gases. In real gases, there are various effects (e.g., van der Waals interactions, vortical flow, relativistic speed limits, and quantumexchange interactions) that can make their speed distribution different from the Maxwell–Boltzmann form. However, rarefied
gases at ordinary temperatures behave very nearly like an ideal gas and
the Maxwell speed distribution is an excellent approximation for such
gases. Thus, it forms the basis of the Kinetic theory of gases, which provides a simplified explanation of many fundamental gaseous properties, including pressure and diffusion.[4]

Distribution function

The speed probability density functions of the speeds of a few noble gases at a temperature of 298.15 K (25 °C). The y-axis
is in s/m so that the area under any section of the curve (which
represents the probability of the speed being in that range) is
dimensionless.

where m{\displaystyle m} is the particle mass and kT{\displaystyle kT} is the product of Boltzmann's constant and thermodynamic temperature.
An interesting point to be noted is that the Maxwell-Boltzmann
distribution will not vary with the value of m/T i.e the ratio of mass
of the molecule to its absolute temperature; mathematically (Derivative
of f(v)/derivative of (m/T))=0. This probability density function gives the probability, per unit speed, of finding the particle with a speed near v{\displaystyle v}. This equation is simply the Maxwell distribution (given in the infobox) with distribution parameter a=kT/m{\displaystyle a={\sqrt {kT/m}}}. In probability theory the Maxwell–Boltzmann distribution is a chi distribution with three degrees of freedom and scale parametera=kT/m{\displaystyle a={\sqrt {kT/m}}}.

Note that a distribution (function) is not the same as the
probability. The distribution (function) stands for an average number,
as in all three kinds of statistics (Maxwell–Boltzmann, Bose–Einstein, Fermi–Dirac). With the Darwin–Fowler method of mean values the Maxwell–Boltzmann distribution is obtained as an exact result.

Typical speeds

The mean speed, most probable speed (mode), and root-mean-square can be obtained from properties of the Maxwell distribution.

The most probable speed, vp, is the speed most likely to be possessed by any molecule (of the same mass m) in the system and corresponds to the maximum value or mode of f(v). To find it, we calculate the derivativedf/dv, set it to zero and solve for v:

Derivation and related distributions

The original derivation in 1860 by James Clerk Maxwell was an argument based on molecular collisions of the Kinetic theory of gases
as well as certain symmetries in the speed distribution function;
Maxwell also gave an early argument that these molecular collisions
entail a tendency towards equilibrium.[2][6] After Maxwell, Ludwig Boltzmann in 1872[7]
also derived the distribution on mechanical grounds and argued that
gases should over time tend toward this distribution, due to collisions
(see H-theorem). He later (1877)[8] derived the distribution again under the framework of statistical thermodynamics. The derivations in this section are along the lines of Boltzmann's 1877 derivation, starting with result known as Maxwell–Boltzmann statistics
(from statistical thermodynamics). Maxwell–Boltzmann statistics gives
the average number of particles found in a given single-particle microstate, under certain assumptions:[1][9]

The assumptions of this equation are that the particles do not
interact, and that they are classical; this means that each particle's
state can be considered independently from the other particles' states. Additionally, the particles are assumed to be in thermal equilibrium.
The denominator in Equation (1) is simply a normalizing factor so that the Ni/N add up to 1 — in other words it is a kind of partition function (for the single-particle system, not the usual partition function of the entire system).

Because velocity and speed are related to energy, Equation (1)
can be used to derive relationships between temperature and the speeds
of gas particles. All that is needed is to discover the density of
microstates in energy, which is determined by dividing up momentum space
into equal sized regions.

The distribution is seen to be the product of three independent normally distributed variables px{\displaystyle p_{x}}, py{\displaystyle p_{y}}, and pz{\displaystyle p_{z}}, with variance mkT{\displaystyle mkT}. Additionally, it can be seen that the magnitude of momentum will be distributed as a Maxwell–Boltzmann distribution, with a=mkT{\displaystyle a={\sqrt {mkT}}}.
The Maxwell–Boltzmann distribution for the momentum (or equally for the
velocities) can be obtained more fundamentally using the H-theorem at equilibrium within the Kinetic theory of gases framework.

Distribution for the energy

where d3p{\displaystyle d^{3}{\textbf {p}}} is the infinitesimal phase-space volume of momenta corresponding to the energy interval dE{\displaystyle dE}. Making use of the spherical symmetry of the energy-momentum dispersion relation E=|p|2/2m{\displaystyle E=|{\textbf {p}}|^{2}/2m}, this can be expressed in terms of dE{\displaystyle dE} as

Since the energy is proportional to the sum of the squares of the
three normally distributed momentum components, this distribution is a gamma distribution; in particular, it is a chi-squared distribution with three degrees of freedom.

By the equipartition theorem,
this energy is evenly distributed among all three degrees of freedom,
so that the energy per degree of freedom is distributed as a chi-squared
distribution with one degree of freedom:[10]

where ϵ{\displaystyle \epsilon }
is the energy per degree of freedom. At equilibrium, this distribution
will hold true for any number of degrees of freedom. For example, if the
particles are rigid mass dipoles of fixed dipole moment, they will have
three translational degrees of freedom and two additional rotational
degrees of freedom. The energy in each degree of freedom will be
described according to the above chi-squared distribution with one
degree of freedom, and the total energy will be distributed according to
a chi-squared distribution with five degrees of freedom. This has
implications in the theory of the specific heat of a gas.

The Maxwell–Boltzmann distribution can also be obtained by considering the gas to be a type of quantum gas for which the approximation ε >> k T may be made.

Distribution for the velocity vector

Recognizing that the velocity probability density fv is proportional to the momentum probability density function by

Like the momentum, this distribution is seen to be the product of three independent normally distributed variables vx{\displaystyle v_{x}}, vy{\displaystyle v_{y}}, and vz{\displaystyle v_{z}}, but with variance kTm{\displaystyle {\frac {kT}{m}}}. It can also be seen that the Maxwell–Boltzmann velocity distribution for the vector velocity [vx, vy, vz] is the product of the distributions for each of the three directions:

where ϕ{\displaystyle \phi } and θ{\displaystyle \theta }
are the "course" (azimuth of the velocity vector) and "path angle"
(elevation angle of the velocity vector). Integration of the normal
probability density function of the velocity, above, over the course
(from 0 to 2π{\displaystyle 2\pi }) and path angle (from 0 to π{\displaystyle \pi }), with substitution of the speed for the sum of the squares of the vector components, yields the speed distribution.

About Me

My formal training is in chemistry. I also read a great deal of physics and biology. In fact I very much enjoy reading in general, mostly science, but also some fiction and history. I also enjoy computer programming and writing. I like hiking and exploring nature. I also enjoy people; not too much in social settings, but one on one; also, people with interesting or "off-beat" minds draw me to them. I also have some interest in Buddhism.

These days I get a lot more information from the internet, primarily through Wiki. Some television, e. g., documentaries, PBS shows like "Nova" and "Nature".

My favorite science writers are Jacob Bronowski ("The Ascent of Man") and Richard Dawkins (his "The Blind Watchmaker" is right up there up Ascent). I also have a favorite writer on Buddhism, Pema Chodron. Favorite films are "Annie Hall" (by Woody Allen), "The Maltese Falcon", "One Flew Over The Cuckoo's Nest", "As Good As It Gets", "Conspiracy Theory", Monty Python's "Search For The Holy Grail" and "Life of Brian", and a few others which I can't think about at the moment.

I love a number of classical works (Beethoven's "Pastoral", "Afternoon Of A Fawn" and "Clair De Lune" by Debussey , Pachelbel's "Canon" come to mind. My favorite piece is probably Gershwin's "Rhapsody in Blue". But I also enjoy a great deal in modern music, including many jazz pieces, folk songs by people like Dylan, Simon and Garfunkel, a hodgepodge of pieces by Crosby, Stills, and Nash, Niel Young, and practically everything the Beatles wrote.

My life over the last few years has been in some disarray, but I am finally "getting it together.". As I am very much into the sciences and writing, I would like to move more in this direction. I also enjoy teaching. As for my political leanings, most people would probably describe as basically liberal, though not extremely so. My religious leanings are to the absolutely none: I've alluded to my interest in Buddhism, but again this is not any supernatural or scientifically untested aspect of it but in the way it provides a powerful philosophy and set of practical, day to day methods of dealing with myself and the other human beings.